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19 Cards in this Set
 Front
 Back
The ________ of a matrix is formed by writing its columna as rows.

transpose
examples: 2 A =8, A^T =[2 8] 

A matrix that has only one column is called a _____ matrix or ______ vector.

column, column
[ 2 ] A =  3  [ 2 ] 

FINDING THE INVERSE OF A MATRIX BY GAUSSJORDAN ELIMINATION

Let A be a square matrix of order n.
1. Write the n*2n matrix that consists of the given matrix A on the left and the n*n identity matrix I on the right to obtain [A : I]. Note that you separate the matrices A and I by a dotted line. This process is called adjoining the matrices A and I. 2. If possible, row reduce A to I using ERO's on the entire matrix [A : I]. The result will be the matrix [I : A^1]. If this is not possible, then A is not invertible. 3. Check your work by multiplying to see that AA^1 = I = A^1*A. 

Two matrices A = [aij] and B = [bij] are ________ if they have the same size (m * n) and aij = bij for 1 <= i <= m and 1 <= j <= n.

equal


Properties of Zero Matrices:
If A is an m*n matrix and c is a scalar, then the following properties are true. 1. A + Omn = ? 2. A + (A) = ? 3. if cA = Omn,, then c = ? or A = ? 
1. = A
2. = Omn 3. = 0 or = Omn 

A matrix that has only one row is called a ______ matrix or ______ vector.

row, row
A = [2 3 4] 

If A is an invertible matrix, then its inverse is ______. The inverse of A is denoted by A^1.

unique


If A = [aij] and B = [bij] are matrices of size m*n, then their _____ is the m*n matrix given by
A + B = [aij + bij] 
sum, The sum of two matrices of different sizes is undefined.
1 2 +  1 3 = 1+1 2+3 =  0 5  0 1 1 2  01 1+2 1 3 

Find the product AB, where
1 3 3 2 A =  4 2 and B =4 1  5 0 
9 1
4 6 15 10 

Real numbers are referred to as ______.

scalars


A matrix that does not have an inverse is called ______ (or _______).

noninvertible
singular 

You can use A to represent the scalar product ______.

(1)A
A  B = A + (B) 

Properties of Matrix Multiplication
If A, B, and C are matrices (with sizes such that the given matrix products are defined) and c is a scalar, then the following properties are true. 1. A(BC) = ? 2. A(B + C) = ? 3. (A + B)C = ? 4. c(AB) = ? 
1. (AB)C (associative property of multiplication)
2. AB + AC (distributive property) 3. AC + BC (distributive property) 4. (cA)B = A(cB) 

If A = [aij] is an m*n matrix and B = [bij] is an n*p matrix, then the product AB is an ________ matrix.

m*p
AB = [cij] where cij = (Summation from k = 1 to n) aik*bkj = ai1*b1j + ai2*b2j +...+ ain*bnj 

An n*n matrix A is ______ (or _______) if there exist an n*n matrix B such that
AB = BA = In where (In) is the identity matrix of order n. The matrix B is called the (multiplicative) _______ of A. 
invertible
nonsingular inverse 

If A = [aij] is an m*n matrix and c is a scalar, then the _________ _______ of A by c is the m*n matrix given by
cA = [caij]. 
scalar multiple


Properties of the Identity Matrix
If A is a matrix of size m*n, then the following properties are true. 1. AIn = ? 2. ImA = ? 
1. = A
2. = A 

Properties of Transpose
If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true. 1. (A^T)^T = ? 2. (A + B)^T = ? 3. (cA)^T = ? 4. (AB)^T = ? 
1. A
2. A^T + B^T (transpose of a sum) 3. c(A^T) (transpose of a scalar multiple) 4. B^T*A^T (transpose of a product) 

Properties of matrix addition and scalar multiplication:
1. A + B = ? 2. A + (B + C) = ? 3. (cd)A = ? 4. 1A = ? 5. c(A + B) = ? 6. (c + d)A = ? 
1. = B + A (commutative property of addition)
2. = (A + B) + C (associative property of addition) 3. = c(dA) 4. = A 5. cA + cB (distributive property) 6. cA + dA (distributive property) 